Can't you borrow a headset for a couple of hours or don't you have one lying around. Even the little earplug types that come with a lot of little machines - as long as they have a 3.5mm plug. That would be the easiest test to see whether we are talking about a hardware or a software problem.
Your friend could have a point too. It could be a driver problem with the HP monitor. Did you look on the HP website whether there was a particular driver required for the monitor.
Another place to look: right click on the speaker icon next to the clock (bottom right) and go to "Playback Devices". See whether your monitor speakers are set as "Default Device" (it would show a checkmark in a green circle). If you right click on any shown device that is not greyed out, you can set that as default. If there are several possible, change them around. Maybe you find the one that fits.